I tested this today. Here were our findings:
The host controls which missions are available or not. (Enemy and weapon levels are based on the zone.) However, those missions can be accomplished by anyone in the host's game. Once completed, that mission is completed for all participating characters, even if they weren't normally able to complete the mission because they hadn't progressed that far in the game in single player yet.
All participating players receive the same XP, rewards, and mission completions for their character.
In other words, if you run with someone who is further ahead in the plot then you, when you leave and go back to your own game (not on the host's co-op server), then you will see any missions completed during that time with the first host, even if you haven't got to that point in the game yet. Then, once you get to those missions that you completed on another host's game, since the missions are already complete for you, you can just go on to the next, uncompleted mission as normal.
As a host, the game progress is the same as if you were playing single player.
The exception to all of this is for the Story Missions. You can complete them with the host, but when you go back to your own, non-host, instance, you will still be at your last uncompleted Story Mission if there was a gap in between the story missions you completed with the host. UPDATE: Once you catch up to completed Story Missions, you get a dialog asking if you'd like to 'fast forward', skipping missions you've already completed in other sessions:
So yes, in my opinion, they "fixed" the ineligible mission message from the first Borderlands.
If you complete a mission, then go back either on your own in single player or as the host, after having already completed it previously in a previous session, you get a message in the reward screen saying something along the lines of 'you have already completed the mission and received your reward' and then don't get the XP or item again — because you already got it the first time you completed it.
Playing through in Vault Hunter Mode doesn't count against this repeat reward limit though; missions there can be completed once more of course before you would get this message again in VHM.